Problem: Al, Betty, and Clare split $\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $\$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $\$100$. What was Al's original portion?
Denote the original portions for Al, Betty, and Clare as $a$, $b$, and $c$, respectively.  Then \[
a + b + c = 1000\quad\text{and}\quad a-100 + 2(b+c) = 1500.
\] Substituting $b+c=1000-a$ in the second equation,   we have \[
a -100 + 2(1000-a)=1500.
\] This yields $a=\boxed{400}$, which is Al's original portion.

Note that although we know that $b+c = 600$, we have no way of determining either $b$ or $c$.